##
**Rational points on elliptic curves.**
*(English)*
Zbl 0752.14034

Undergraduate Texts in Mathematics. New York: Springer-Verlag. x, 281 p. (1992).

The book gives a good introduction for students which are interested in Diophantine equations and arithmetic geometry. It is based on lectures of J. Tate from 1961. It contains a lot of exercises. Often further developments and applications are explained, for instance Lenstra’s algorithm for factorisation of integers using elliptic curves.

The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. — At the end one finds complex multiplication and Galois representations associated to torsion points.

The algebraic geometry needed for the purpose of the book (for instance Bézout’s theorem) is presented in an appendix.

The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. — At the end one finds complex multiplication and Galois representations associated to torsion points.

The algebraic geometry needed for the purpose of the book (for instance Bézout’s theorem) is presented in an appendix.

Reviewer: Gerhard Pfister (Berlin)

### MSC:

11G05 | Elliptic curves over global fields |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

11G15 | Complex multiplication and moduli of abelian varieties |

11D25 | Cubic and quartic Diophantine equations |

14G05 | Rational points |

14G15 | Finite ground fields in algebraic geometry |

14H52 | Elliptic curves |

### Keywords:

Diophantine equations; elliptic curves; Nagell-Lutz theorem; Mordell-Weil theorem; rational points; Thue-Siegel theorem; integer points; finite fields; complex multiplication; torsion points
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\textit{J. H. Silverman} and \textit{J. Tate}, Rational points on elliptic curves. New York: Springer-Verlag (1992; Zbl 0752.14034)

### Digital Library of Mathematical Functions:

§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem ‣ §22.18 Mathematical Applications ‣ Applications ‣ Chapter 22 Jacobian Elliptic FunctionsOther Notations ‣ §23.1 Special Notation ‣ Notation ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(ii) Elliptic Curves ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions

§23.20(v) Modular Functions and Number Theory ‣ §23.20 Mathematical Applications ‣ Applications ‣ Chapter 23 Weierstrass Elliptic and Modular Functions